Contact handles, duality, and sutured Floer homology

TL;DR

This paper provides explicit contact handle constructions for sutured Floer homology, establishes a duality for sutured manifold cobordisms, and confirms the equivalence of two link cobordism map definitions in link Floer homology.

Contribution

It introduces explicit contact handle models for gluing maps, proves a duality result for sutured manifold cobordisms, and shows the consistency of different link cobordism map constructions.

Findings

Explicit construction of gluing maps via contact handles

Duality result for sutured manifold cobordisms

Equivalence of link cobordism maps in link Floer homology

Abstract

We give an explicit construction of the Honda--Kazez--Mati\'c gluing maps in terms of contact handles. We use this to prove a duality result for turning a sutured manifold cobordism around, and to compute the trace in the sutured Floer TQFT. We also show that the decorated link cobordism maps on the hat version of link Floer homology defined by the first author via sutured manifold cobordisms and by the second author via elementary cobordisms agree.